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Despite the recent success of deep learning for time series forecasting, these methods are not scalable for many real-world applications where data arrives sequentially. Training deep neural forecasters on the fly is notoriously challenging because of their limited ability to adapt to non-stationary environments and remember old knowledge. We argue that the fast adaptation capability of deep neural networks is critical and successful solutions require handling changes to both new and recurring patterns effectively. In this work, inspired by the Complementary Learning Systems (CLS) theory, we propose Fast and Slow learning Network (FSNet) as a novel framework to address the challenges of online forecasting. Particularly, FSNet improves the slowly-learned backbone by dynamically balancing fast adaptation to recent changes and retrieving similar old knowledge. FSNet achieves this mechanism via an interaction between two novel complementary components: (i) a per-layer adapter to support fast learning from individual layers, and (ii) an associative memory to support remembering, updating, and recalling repeating events. Extensive experiments on real and synthetic datasets validate FSNet's efficacy and robustness to both new and recurring patterns. Our code is publicly available at: \url{https://github.com/salesforce/fsnet/}.